2. All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.
3. Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, … Rational numbers can be expressed in the form , where a and b are both integers and b ≠ 0: , , . a b 1 2 7 1 9 10
4. Terminating decimals are rational numbers in decimal form that have a finite number of digits: 1.5, 2.75, 4.0 Repeating decimals are rational numbers in decimal form that have a block of one or more digits that repeat continuously: 1.3, 0.6, 2.14 Irrational numbers cannot be expressed in the form . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: , , a b
5. Example 3: Classifying Real Numbers Write all classifications that apply to each Real number. A. –32 – 32 = – = –32.0 32 1 32 can be written as a fraction and a decimal. rational number, integer, terminating decimal B. 5 5 = = 5.0 5 1 5 can be written as a fraction and a decimal. rational number, integer, whole number, natural number, terminating decimal
6. Write all classifications that apply to each real number. rational number, repeating decimal 3b. –12 rational number, terminating decimal, integer irrational number Check It Out! Example 3 32 can be written as a fraction and a decimal. The digits continue with no pattern. 3a. 7 4 9 3c. 67 9 = 7.444… = 7.4 7 can be written as a repeating decimal. 49 – 12 = – = –12.0 12 1 = 3.16227766…
7. A variable is a letter or a symbol used to represent a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic expression may contain variables, constants, and operations.
8. You will need to translate between algebraic expressions and words to be successful in math. Plus, sum, increased by Minus, difference, less than Times, product, equal groups of Divided by, quotient – +
9. These expressions all mean “2 times y ”: 2 y 2( y ) 2• y (2)( y ) 2 x y (2) y Writing Math
10. Give two ways to write each algebra expression in words. A. 9 + r B. q – 3 the sum of 9 and r 9 increased by r the product of m and 7 m times 7 the difference of q and 3 3 less than q the quotient of j and 6 j divided by 6 Example 1: Translating from Algebra to Words C. 7 m D. j 6
11. To translate words into algebraic expressions, look for words that indicate the action that is taking place. Put together, combine Find how much more or less Put together equal groups Separate into equal groups Divide Multiply Subtract Add
12. John types 62 words per minute. Write an expression for the number of words he types in m minutes. m represents the number of minutes that John types. 62 · m or 62 m Think: m groups of 62 words Example 2A: Translating from Words to Algebra
13. Roberto is 4 years older than Emily, who is y years old. Write an expression for Roberto’s age y represents Emily’s age. y + 4 Think: “older than” means “greater than.” Example 2B: Translating from Words to Algebra
14. Joey earns $5 for each car he washes. Write an expression for the number of cars Joey must wash to earn d dollars. d represents the total amount that Joey will earn. Think: How many groups of $5 are in d? Example 2C: Translating from Words to Algebra
15. Lou drives at 65 mi/h. Write an expression for the number of miles that Lou drives in t hours. Check It Out! Example 2a t represents the number of hours that Lou drives. 65 t Think: number of hours times rate per hour.
16. Miriam is 5 cm taller than her sister, than her sister who is m centimeters tall. Write an expression for Miriam’s height in centimeters. Check It Out! Example 2b m represents Miriam’s sister's height in centimeters. m + 5 Think: Miriam's height is 5 added to her sister's height.
17. Check It Out! Example 2c d represents the amount of money Elaine will earn each day. 32 d Think: The number of days times the amount Elaine would earn each day. Elaine earns $32 per day. Write an expression for the amount she earns in d days.
18. Give two ways to write each algebraic expression in words. 1. j – 3 2. 4 p 3. Mark is 5 years older than Juan, who is y years old. Write an expression for Mark’s age. The difference of j and 3; 3 less than j. 4 times p ; The product of 4 and p. y + 5 Lesson Quiz: Part I
